temperature = 300 K, that is,
<<
so virtually all donors are posi- tively ionized and virtually all acceptors are negatively ionized at room temperature. The donor and acceptor atoms that we have been discussing are known as shallowcenters, that is, shallow traps of electrons or holes, because their excitation energies are much less than that of the bandgap
<<
There are other centers with energy levels that lie deep within the forbidden gap, often closer to its center than to the top or bottom, in contrast to the case with shallow donors and acceptors. Since generally>>
these traps are not extensively ionized, and the energies involved in exciting or ionizing them are not small. Examples of deep centers are defects associated with broken bonds, or strain involving displacements of atoms. In Chapter 8 we discuss how deep centers can produce characteristic optical spectro- scopic effects.2.3.2.
MobilityAnother important parameter of a semiconductor is the mobility or charge carrier velocity per unit electric field E, given by the expression This parameter is defined as positive for both electrons and holes. Table B.9 lists the mobilities and for electrons and holes, respectively, in the semiconductors under consideration. The electrical conductivity is the sum of contributions from the concentrations of electrons n and of holes p in accordance with the expression
where e is the electronic charge. The mobilities have a weak power-law temperature dependence T", and the pronounced T dependence of the conductivity is due principally to the dependence of the electron and hole concentrations on the tem- perature. In doped semiconductors this generally arises mainly from the Boltzmann factor associated with the ionization energies of the donors or acceptors. Typical ionization energies for donors and acceptors in Si and Ge listed in Table B. 10 are in the range from 0.0096 to 0.16 which is much less than the bandgap energies 1.1 and 0.66 of Si and Ge, respectively. Figure 2.12 shows the locations of donor and acceptor levels on an energy band plot, and makes clear that their respective ionization energies are much less that The thermal energy = 0.026 at room temperature (300 K) often comparable to the ionization energies. In intrinsic or undoped materials the main contribution is the exponential factor in the following expression from the law of mass action
= (2.15)
where the intrinsic concentrations of electrons and holes are equal to each other because the thermal excitation of electrons to the conduction band leaves behind
the same number of holes in the valence band, that is We see that the expression (2.15) contains the product of the effective masses and of the electrons and holes, respectively, and the ratios and of these effective masses to the free-electron mass are presented in Table B.8. These effective masses strongly influence the properties of excitons to be discussed next.
2.3.3. Excitons
An ordinary negative electron and a positive electron, called a positron, situated a distance apart in free space experience an attractive force called the Coulomb force,
which has the value where e is their charge and is the dielectric constant of free space. A quantum-mechanical calculation shows that the electron and positron interact to form an atom called positronium which has bound-state energies given by the Rydberg formula introduced by Niels Bohr in 19 13 to explain the hydrogen atom
(2.16)
where is the Bohr radius given by = = 0.0529 is the electron (and positron) mass, and the quantum number takes on the values
n = .
.
, For the lowest energy or ground state, which has = 1, the energy is 6.8 which is exactly half the ground-state energy of a hydrogen atom, since the effective mass of the bound electron-positron pair is half of that of the bound electron-proton pair in the hydrogen atom. Figure 2.20 shows the energy levels of positronium as a of the quantum number n. This set of energy levels is referred to as a Rydberg series. The continuum at the top of the figure is the region of positive energies where the electron and hole are so far away each other that the Coulomb interaction no longer has an appreciable effect, and the energy is all of the kinetic type, = or energy of motion, where is the velocity a n d p = mu is the momentum.The analog of positronium in a solid such as a semiconductor is the bound state of an electron-hole pair, called an exciton. For a semiconductor the electron is in the conduction band, and the hole is in the valence band. The electron and hole both have effective masses and respectively, which are less than that of a free electron, so the effective mass is given by =
+
When the electron effective mass is appreciably less than the hole effective mass,<<
the relationship between them is conveniently written in the form(2.17)
which shows that for this case becomes comparable with the electron mass. For example, if = then = A comparison of the data in Table B.8 shows that this situation is typical for GaAs-type semiconductors. We also see from
2.3. LOCALIZED PARTICLES 1 10 100
E
W 1 10 n = 4 n = 3 n = 2 n = l Exciton Positronium Hydrogen n = 5 n = 4 n = 3 n = 4 n = 2 n = 3 n = 2 n = l n = lFigure 2.20. First few energy levels in the Rydberg series of a hydrogen atom (left), positronium (center), and a typical exciton (right).
Table B.ll that the relative dielectric constant has the range of values 7.2 17.7 for these materials, where is the dielectric constant of free space. Both of these factors have the effect of decreasing the exciton energy E,, from that of positronium, and as a result this energy is given by
1 e’
E,, - (2.18)
as shown plotted in Fig. 2.20. These same two factors also increase the effective Bohr radius of the electron orbit, and it becomes
Using the GaAs electron effective mass and the heavy-hole effective mass values from Table B.8, Eq. (2.17) gives = 0.059. Utilizing the dielectric constant value from Table B . l l , we obtain, with the aid of Eqs. (2.18) and for GaAs (2.20) where is the ground-state (n = 1) energy. This demonstrates that an exciton extends over quite a few atoms of the lattice, and its radius in GaAs is comparable with the dimensions of a typical nanostructure. A n exciton has the properties of a particle; it is mobile and able to move around the lattice. It also exhibits characteristic optical spectra. Figure 2.20 plots the energy levels for an exciton with the ground-state energy = 18
Technically speaking, the exciton that we have just discussed is a weakly bound electron-hole pair called a exciton. A strongly or tightly bound exciton, called a exciton, is similar to a long-lived excited state of an atom or a molecule. It is also mobile, and can move around the lattice by the transfer of the excitation or excited-state charge between adjacent atoms or molecules. Almost all the excitons encountered in semiconductors and in are of the Mott-Wannier type, so they are the only ones discussed in this book.